Geometric limits of knot complements

TL;DR

This paper demonstrates that certain hyperbolic 3-manifolds with specific topological properties can be approximated as geometric limits of hyperbolic knot complements in the 3-sphere, revealing new insights into their geometric structure.

Contribution

It establishes conditions under which hyperbolic 3-manifolds are limits of knot complements and shows limitations for manifolds with two convex cocompact ends.

Findings

Existence of hyperbolic knot complements with arbitrarily large embedded balls.

Certain hyperbolic 3-manifolds are geometric limits of knot complements.

Manifolds with two convex cocompact ends cannot be limits of knot complements.

Abstract

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3--manifold with two convex cocompact ends cannot be a geometric limit of knot complements in $\BS^3$.